I don't know that testing at the first and 13th fret is going to tell you the whole story on neck relief.

how much relief do you see if you fret at the first fret and say the 19th or 20th on the B string side? (what bongolation said)

if there's not much relief and the saddles are already set low, then you will probably need to shim the neck (like you said). if you're knowledgeable with working with shims to create neck angle you can probably get that thing playing nice assuming there's no twist in the neck and the truss rod still works well.